@donothingsuccessfully - The problem with that is that there's a Popeye cartoon from 16 years before that movie named "Shiver me Timbers". At best Mr. Newton tweaked and popularized an existing cultural meme.
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T.E.D.Nov 21 '13 at 15:17

Notice in particular the unstressed pronunciation of my as /mɪ/ listed above.

In other words, unstressed my as a possessive adjective was once, and to some extent still is, pronounced as though it were me. It simply means my when so used, and is sometimes still used in reported speech to represent dialect pronunciation.

The "bloody" I know not of, in this context at least [ :-) ] , but the use of 'me' for 'my' in Pirate's putative Parlance is very time hounoured - dating back to at least my childhood, which is probably more distance than that of the childhood of the average list member - and probably a century or two prior to that.

Searching for "me" when used to mean "my" is a challenge which Google is not well designed to rise to, but use of a common (allegedly) pirate's term will suffice.

"Shiver me timbers" - 1,090,000 hits

"Shiver my timbers" - 89,500 hits

While a sample size of one falls short of the test for being definitive (by a few orders of magnitude) it is a good demonstration of the universality of the term.

The first popularized pirate tale was Robert Louis Stevenson's Treasure Island in 1883. However, Mr. Stevenson's pirates all used proper Victorian English(!), like "Shiver my timbers." So no possessive "me"s were used there.

We do know that by the 1930's this bit of canonical pirate lingo had developed, as both Errol Flynn's 1935 Captain Blood and the 1934 Popeye cartoon "Shiver Me Timbers!" used it.

Wikipedia claims that the possessive me is used in many "nonstandard" British dialects. If so, the early use of this form was likely an attempt by writers to show that the typical pirate came from a lower-class and/or rural British background.

They further speculate that possessive "me" in those dialects might have derived from the way "my" was pronounced before the great vowel shift. In Middle English, "my" before a consonant was indeed pronounced just like the modern "me", while "me" would have been pronounced similar to the modern "may".